Cubic surfaces and configurations of 6 points

A smooth cubic surface $$X\subset \mathbb{P}^3$$ is isomorphic to $$\mathbb{P}^2$$ blown up at six points, so there should be a rational map

$${\rm Hilb}^6\mathbb{P}^2\dashrightarrow H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4$$

Given a 1-parameter family $$Z_t$$ of length six subschemes of $$\mathbb{P}^2$$, where $$Z_t$$ is reduced for $$t\neq 0$$, specializing to $$Z_0$$, what happens to the singularities of the corresponding 1-parameter family $$X_t$$ of cubic surfaces? For example, I know that $$X_0$$ having an $$A_1$$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.

• A cubic surface is a blowup in many different ways. So, the map is in the opposite direction. Dec 11 '18 at 18:21
• Moreover this map is not birational, again for the reasons which Sasha states Dec 11 '18 at 19:14
• I'm sorry for the mistake, thanks for the correction. Dec 11 '18 at 19:24
• The Séminaire sur les singularités des surfaces (Springer Lecture Notes 777) contains a detailed study of all rational singularities which can occur in that way.
– abx
Dec 11 '18 at 19:32

Any smooth cubic surface of $$\mathbb{P}^3$$ is the blow-up of six points of $$\mathbb{P}^2$$ such that no three are collinear and no six are on a conic. More generally, if $$\sigma\colon S\to \mathbb{P}^2$$ is the blow-up of six points, maybe some infinitely near, such that $$−K_S$$ is nef (or equivalently such that all curves of negative self-intersection are smooth rational (−1)-curves or (−2)-curves), the anticanonical map $$\eta\colon S \to\mathbb{P}^3$$ is a birational morphism to a normal cubic surface, which contracts all $$(−2)$$- curves of $$S$$ (it is an isomorphism if and only if $$−K_S$$ is ample, which means that $$S$$ is del Pezzo).
Conversely, all normal cubic surfaces except cones over smooth cubic curves are obtained in this way. Indeed such a surface $$S$$ admits only double points, is rational (project from a singularity) and satisfies $$H^1(S, \mathcal{O}_S) = H^2(S, \mathcal{O}_S) = 0$$ (use the exact sequence $$0 \to\mathcal{O}(−3) → \mathcal{O} → \mathcal{O}(S) → 0$$); hence by [ I. Dolgachev, Classical algebraic geometry: a modern view (Cambridge University Press, Cambridge, 2012, Proposition 8.1.8(ii)] $$S$$ admits only rational double points. Alternatively, one can use the classification of singularities on cubic surfaces in [J. W. Bruce and C. T. C. Wall, ‘On the classification of cubic surfaces’, J. London Math. Soc. (2) 19 (1979) 245–256., § 2], which does not rely on a cohomological argument. Then the minimal resolution $$\hat{S} \to S$$ is a weak del Pezzo surface and is the blow-up of six points of $$\mathbb{P}^2$$ by [Dolgachev (as above), Theorem 8.1.13]. The singularities you can obtain follow from this description. You can for instance get $$A_1$$, $$\ldots$$, $$A_5$$, $$A_6$$, $$D_5$$.